The Harsanyi dividend is defined as follows:
$d_v (S) = \sum_{R \subseteq S} (-1)^{|S|-|R|} v(R)$
Supermodularity is defined as follows, for $S \subseteq T \subseteq N$: $v(S \cup \{i\}) - v(S) \leq v(T \cup \{i\}) - v(T)$.
It is easy to prove that positive Harsanyi dividends imply supermodularity. The implication the other way is not true in general, since we can design a supermodular game that has a negative dividend. Therefore, I am trying to find a sufficient extra condition such that we can prove that Harsanyi dividends are positive:
Supermodularity + ??? $\implies d_v (S) \geq 0 \ \forall S \subseteq N$.
I have no luck finding this condition. I already tried writing down the proof, hoping that I would get the right insight when I got stuck, but that is not happening yet:
$d_v(S) = \sum_{R \subseteq S} (-1)^{|S|-|R|} v(R) = \sum_{R \subseteq S \backslash \{i,j\}} (-1)^{|S|-|R|}(v(R \cup \{i,j\}) - v(R \cup \{i\}) - v(R \cup \{j\}) + v(R)) = ... $
I am not sure that last step is useful, but this is how far I got. Can anyone please help me finding the wanted condition?
